Geometric Spin-Orbit Coupling Resolves the Contradictory CISS Effect in Chiral Single Molecules

Technical Summary: Geometric Spin-Orbit Coupling Resolves the Contradictory CISS Effect in Chiral Single Molecules

Problem Statement
The Chirality-Induced Spin Selectivity (CISS) effect, where chiral materials filter electron spins without external magnetic fields, has been widely observed in macroscopic systems and large biomolecules. However, a significant contradiction exists regarding its manifestation in chiral single molecules (CSMs). While some studies report high spin polarization (SP) up to 40% in four specific classes of CSMs (atropisomers, enantiomers, bowl-shaped derivatives, and dipolar L-histidine), a recent high-precision study involving nearly a thousand individual tests on the same systems failed to detect significant CISS signals. This discrepancy challenges the universality of the CISS effect at the single-molecule level. Existing theoretical frameworks, primarily based on atomic intrinsic spin-orbit coupling (SOC) or effective electric field-driven SOC, struggle to explain high SP in light-element organic molecules or to account for CISS in irregular, non-helical chiral geometries. Furthermore, these models often fail to reconcile why identical molecules yield contradictory results under different experimental conditions.

Methodology
To resolve these inconsistencies, the authors developed a unified theoretical framework incorporating geometric spin-orbit coupling (SOC) and environmental decoherence.

1. Geometric SOC Model: Instead of relying on atomic intrinsic SOC (which is weak in light atoms) or external field gradients, the authors propose that SOC arises from the relativistic effect of electrons moving along curved or twisted pathways within the chiral molecule. This "geometric SOC" is linked to the local spatial curvature ($\kappa$) and the accumulated Berry phase. The strength of this coupling is derived from the Frenet-Serret frame of the molecular path, where the effective acceleration of the electron in a non-inertial frame induces a momentum-dependent magnetic field.
2. Structural Modeling: The study utilizes a tight-binding model to simulate three distinct chiral chain geometries:
   • Model A: A cone-shaped helix (radius decreases along the axis).
   • Model B: A uniform helix (constant radius).
   • Model C: A progressive-pitch helix (pitch increases along the axis).
     These models allow for a continuous transition from regular helices to irregular, aperiodic curved arcs.
3. Decoherence and Many-Body Effects: The framework employs the Büttiker virtual-electrode technique to simulate environmental decoherence ($\Gamma_d$), representing thermal vibrations that destroy electron phase memory. The study further incorporates many-body correlations:
   • Electron-Electron (e-e) Correlation: Modeled via on-site Coulomb repulsion ($U$).
   • Electron-Vibration (e-vib) Coupling: Modeled by expanding the local curvature around equilibrium positions to include atomic displacement vectors.
4. Calculation: Spin-up and spin-down conductances ($G_{\uparrow/\downarrow}$) are calculated using the Landauer-Büttiker formula combined with Green's function theory to determine SP ($SP = (G_{\uparrow} - G_{\downarrow}) / (G_{\uparrow} + G_{\downarrow})$).

Key Results

• Non-Monotonic Dependence on Decoherence: The most critical finding is that the CISS effect is suppressed in both the strong-coherence and strong-decoherence regimes. Significant spin polarization emerges only in an intermediate-decoherence regime.
  • Strong Coherence: Corresponds to highly coherent transport (e.g., specific STM break-junction setups), where SP is negligible (<1%), explaining the null results in recent high-precision experiments.
  • Intermediate Decoherence: Corresponds to typical room-temperature conditions with moderate scattering, where SP can reach 40%, aligning with positive experimental reports.
• Geometric Evolution: The framework successfully describes the evolution of CISS from regular helical symmetry to irregular (point-symmetric or axially symmetric) chirality. Even molecules that do not complete a full helical turn can exhibit observable CISS if they possess sufficient curvature and fall within the intermediate decoherence window.
• Role of Many-Body Interactions:
  • Electron-Electron Correlation: Enhances SP significantly, particularly in larger molecules and in the strong-coherence regime where single-particle CISS would otherwise be absent. It creates resonance peaks and widens the energy window for observable SP.
  • Electron-Vibration Coupling: Also enhances SP but is more effective in smaller molecules. It acts as an intrinsic decoherence source that, synergistically with external decoherence, stabilizes spin-polarized channels.
• Temperature Dependence: Increasing temperature enhances spin polarization, particularly when mediated by electron-vibration coupling, consistent with experimental observations of robust CISS at room temperature.

Significance and Claims
The paper claims to provide a unified theoretical framework that resolves the contradictory experimental observations regarding CISS in single molecules. By shifting the focus from atomic intrinsic SOC to geometric SOC and explicitly accounting for the decoherence strength, the authors demonstrate that the presence or absence of CISS is not a contradiction of the effect's existence but a consequence of the specific experimental regime (coherence length) and molecular geometry.

The study asserts that:

1. The CISS effect is universal across single-molecule systems, provided the decoherence strength lies within an intermediate range (specifically predicting optimal observation at $\Gamma_d \approx 0.005t_1$, corresponding to a coherence length of $\sim 10$ nm at room temperature).
2. Geometric SOC is the dominant mechanism for high SP in light-element chiral molecules, capable of exceeding atomic intrinsic SOC by one to two orders of magnitude.
3. The framework extends the applicability of CISS theory beyond regular helices to irregular and non-helical chiral structures, addressing a long-standing gap in the field.
4. The interplay between decoherence, molecular size, and many-body interactions (e-e and e-vib) dictates the observability of the effect, offering a physical explanation for why different experimental setups yield divergent results on the same molecular systems.

This work establishes a basis for unifying CISS phenomena and provides concrete guidelines for designing future single-molecule experiments to reliably detect spin selectivity.